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<title>Figure 8.9: Robust linear discrimination problem</title>
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<h1>Figure 8.9: Robust linear discrimination problem</h1>
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<pre class="codeinput">
<span class="comment">% Section 8.6.1, Boyd &amp; Vandenberghe "Convex Optimization"</span>
<span class="comment">% Original by Lieven Vandenberghe</span>
<span class="comment">% Adapted for CVX by Joelle Skaf - 10/16/05</span>
<span class="comment">% (a figure is generated)</span>
<span class="comment">%</span>
<span class="comment">% The goal is to find a function f(x) = a'*x - b that classifies the points</span>
<span class="comment">% {x_1,...,x_N} and {y_1,...,y_M} with maximal 'gap'. a and b can be</span>
<span class="comment">% obtained by solving the following problem:</span>
<span class="comment">%           maximize    t</span>
<span class="comment">%               s.t.    a'*x_i - b &gt;=  t     for i = 1,...,N</span>
<span class="comment">%                       a'*y_i - b &lt;= -t     for i = 1,...,M</span>
<span class="comment">%                       ||a||_2 &lt;= 1</span>

<span class="comment">% data generation</span>
n = 2;
randn(<span class="string">'state'</span>,3);
N = 10; M = 6;
Y = [1.5+1*randn(1,M); 2*randn(1,M)];
X = [-1.5+1*randn(1,N); 2*randn(1,N)];
T = [-1 1; 1 1];
Y = T*Y;  X = T*X;

<span class="comment">% Solution via CVX</span>
cvx_begin
    variables <span class="string">a(n)</span> <span class="string">b(1)</span> <span class="string">t(1)</span>
    maximize (t)
    X'*a - b &gt;= t;
    Y'*a - b &lt;= -t;
    norm(a) &lt;= 1;
cvx_end

<span class="comment">% Displaying results</span>
linewidth = 0.5;  <span class="comment">% for the squares and circles</span>
t_min = min([X(1,:),Y(1,:)]);
t_max = max([X(1,:),Y(1,:)]);
tt = linspace(t_min-1,t_max+1,100);
p = -a(1)*tt/a(2) + b/a(2);
p1 = -a(1)*tt/a(2) + (b+t)/a(2);
p2 = -a(1)*tt/a(2) + (b-t)/a(2);

graph = plot(X(1,:),X(2,:), <span class="string">'o'</span>, Y(1,:), Y(2,:), <span class="string">'o'</span>);
set(graph(1),<span class="string">'LineWidth'</span>,linewidth);
set(graph(2),<span class="string">'LineWidth'</span>,linewidth);
set(graph(2),<span class="string">'MarkerFaceColor'</span>,[0 0.5 0]);
hold <span class="string">on</span>;
plot(tt,p, <span class="string">'-r'</span>, tt,p1, <span class="string">'--r'</span>, tt,p2, <span class="string">'--r'</span>);
axis <span class="string">equal</span>
title(<span class="string">'Robust linear discrimination problem'</span>);
<span class="comment">% print -deps linsep.eps</span>
</pre>
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<pre class="codeoutput">
 
Calling sedumi: 20 variables, 5 equality constraints
   For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 5, order n = 20, dim = 21, blocks = 2
nnz(A) = 68 + 0, nnz(ADA) = 21, nnz(L) = 13
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            4.69E+01 0.000
  1 :  -3.49E-01 1.71E+01 0.000 0.3637 0.9000 0.9000   2.59  1  1  9.9E+00
  2 :   7.65E-03 5.42E+00 0.000 0.3177 0.9000 0.9000   2.36  1  1  2.4E+00
  3 :   2.62E-01 1.27E+00 0.000 0.2336 0.9000 0.9000   0.23  1  1  1.0E+00
  4 :   4.73E-01 3.00E-01 0.000 0.2372 0.9000 0.9000   0.58  1  1  2.7E-01
  5 :   5.05E-01 2.91E-02 0.000 0.0969 0.9900 0.9900   0.98  1  1  2.7E-02
  6 :   5.11E-01 1.39E-05 0.000 0.0005 0.9999 0.9999   1.00  1  1  1.5E-05
  7 :   5.11E-01 1.04E-06 0.290 0.0748 0.9900 0.9900   1.00  1  1  1.1E-06
  8 :   5.11E-01 1.96E-07 0.000 0.1874 0.9000 0.9063   1.00  1  1  2.2E-07
  9 :   5.11E-01 2.87E-09 0.000 0.0147 0.9901 0.9900   1.00  1  1  3.3E-09

iter seconds digits       c*x               b*y
  9      0.0   8.6  5.1122989899e-01  5.1122989774e-01
|Ax-b| =   1.6e-09, [Ay-c]_+ =   3.3E-11, |x|=  1.1e+00, |y|=  1.2e+00

Detailed timing (sec)
   Pre          IPM          Post
0.000E+00    5.000E-02    0.000E+00    
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.51123
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